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Mirrors > Home > ILE Home > Th. List > lereci | GIF version |
Description: The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999.) |
Ref | Expression |
---|---|
ltplus1.1 | ⊢ 𝐴 ∈ ℝ |
prodgt0.2 | ⊢ 𝐵 ∈ ℝ |
Ref | Expression |
---|---|
lereci | ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → (𝐴 ≤ 𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltplus1.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
2 | prodgt0.2 | . . 3 ⊢ 𝐵 ∈ ℝ | |
3 | lerec 8870 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 ≤ 𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴))) | |
4 | 2, 3 | mpanr1 437 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ 0 < 𝐵) → (𝐴 ≤ 𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴))) |
5 | 1, 4 | mpanl1 434 | 1 ⊢ ((0 < 𝐴 ∧ 0 < 𝐵) → (𝐴 ≤ 𝐵 ↔ (1 / 𝐵) ≤ (1 / 𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2160 class class class wbr 4018 (class class class)co 5895 ℝcr 7839 0cc0 7840 1c1 7841 < clt 8021 ≤ cle 8022 / cdiv 8658 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulrcl 7939 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-precex 7950 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-apti 7955 ax-pre-ltadd 7956 ax-pre-mulgt0 7957 ax-pre-mulext 7958 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-reap 8561 df-ap 8568 df-div 8659 |
This theorem is referenced by: (None) |
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